A054493 A Pellian-related recursive sequence.

1, 7, 36, 175, 841, 4032, 19321, 92575, 443556, 2125207, 10182481, 48787200, 233753521, 1119980407, 5366148516, 25710762175, 123187662361, 590227549632, 2827950085801, 13549522879375, 64919664311076, 311048798676007, 1490324329068961, 7140572846668800

Offset

0,2

Comments

This is the r=7 member in the r-family of sequences S_r(n+1) defined in A092184 where more information can be found.

Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 7, P2 = 10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.

Peter Bala, Linear divisibility sequences and Chebyshev polynomials

S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.

Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.

R. Stephan, Boring proof of a nonlinearity

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (6,-6,1)

Formula

a(n) = 5*a(n-1) - a(n-2) + 2, a(0)=1, a(1)=7.

A004254 = sqrt{21*(A054493)^2+28*(A054493)}/7. - James A. Sellers, May 10 2000

a(n) = (1/3)*(-2 + ((5+sqrt(21))/2)^n + ((5-sqrt(21))/2)^n). - Ralf Stephan, Apr 14 2004

G.f.: (1+x)/((1-x)*(1 - 5*x + x^2)) = (1+x)/(1 - 6*x + 6*x^2 - x^3). From the R. Stephan link.

a(n) = 6*a(n-1) - 6*a(n-2) + a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=7.

a(n) = (2*T(n, 5/2)-2)/3, with twice the Chebyshev polynomials of the first kind, 2*T(n, x=5/2)=A003501(n).

a(n) = b(n) + b(n-1), n>=1, with b(n)=A089817(n) the partial sums of S(n, 5)= U(n, 5/2)=A004254(n+1), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind.

From Peter Bala, Mar 25 2014: (Start)

The following formulas assume an offset of 1.

Let {u(n)} be the Lucas sequence in the quadratic integer ring Z[sqrt(7)] defined by the recurrence u(0) = 0, u(1) = 1 and u(n) = sqrt(7)*u(n-1) - u(n-2) for n >= 2. Then a(n) = u(n)^2.

Equivalently, a(n) = U(n-1,sqrt(7)/2)^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = 1/3*( ((sqrt(7) + sqrt(3))/2)^n - ((sqrt(7) - sqrt(3))/2)^n )^2.

a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -5/2; 1, 7/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.

See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

a(2*n - 1) = 7 * A004254(n)^2, a(2*n) = A030221(n)^2 for all n in Z. - Michael Somos, Jan 22 2017

a(n) = a(-2-n) for all n in Z. - Michael Somos, Jan 22 2017

0 = 1 + a(n)*(-2 + a(n) - 5*a(n+1)) + a(n+1)*(-2 + a(n+1)) for all n in Z. - Michael Somos, Jan 22 2017

Example

G.f. = 1 + 7*x + 36*x^2 + 175*x^3 + 841*x^4 + 4032*x^5 + 19321*x^6 + ...

Maple

A054493 := proc(n)
    option remember;
    if n <= 1 then
        6*n+1 ;
    else
        5*procname(n-1)-procname(n-2)+2 ;
    end if ;
end proc:
seq(A054493(n), n=0..10) ; # R. J. Mathar, Apr 16 2018

Mathematica

LinearRecurrence[{6, -6, 1}, {1, 7, 36}, 30] (* Harvey P. Dale, Apr 15 2015 *)
a[ n_] := ChebyshevU[n, Sqrt[7]/2]^2; (* Michael Somos, Jan 22 2017 *)

Prog

(PARI) {a(n) = simplify(polchebyshev(n, 2, quadgen(28)/2)^2)}; /* Michael Somos, Jan 22 2017 */

Crossrefs

Cf. A004254, A100047, A030221 (first differences).

Sequence in context: A058681 A246417 A110310 * A368575 A037538 A037482

Adjacent sequences: A054490 A054491 A054492 * A054494 A054495 A054496

Keyword

easy,nonn

Author

Barry E. Williams, May 06 2000

Extensions

Chebyshev comments from Wolfdieter Lang, Sep 10 2004

Status

approved